The Physicalization of Metamathematics and Its Implications for the Foundations of Mathematics
Abstract
Both metamathematics and physics are posited to emerge from samplings by observers of the unique ruliad structure that corresponds to the entangled limit of all possible computations. The possibility of higher-level mathematics accessible to humans is posited to be the analog for mathematical observers of the perception of physical space for physical observers. A physicalized analysis is given of the bulk limit of traditional axiomatic approaches to the foundations of mathematics, together with explicit empirical metamathematics of some examples of formalized mathematics. General physicalized laws of mathematics are discussed, associated with concepts such as metamathematical motion, inevitable dualities, proof topology and metamathematical singularities. It is argued that mathematics as currently practiced can be viewed as derived from the ruliad in a direct Platonic fashion analogous to our experience of the physical world, and that axiomatic formulation, while often convenient, does not capture the ultimate character of mathematics. Among the implications of this view is that only certain collections of axioms may be consistent with inevitable features of human mathematical observers. A discussion is included of historical and philosophical connections, as well as of foundational implications for the future of mathematics.
Mathematics and Physics Have the Same Foundations
The Underlying Structure of Mathematics and Physics
The Metamodeling of Axiomatic Mathematics
Some Simple Examples with Mathematical Interpretations
Metamathematical Space
The Issue of Generated Variables
Rules Applied to Rules
Accumulative Evolution
Accumulative String Systems
The Case of Hypergraphs
Proofs in Accumulative Systems
Beyond Substitution: Cosubstitution and Bisubstitution
Some First Metamathematical Phenomenology
Relations to Automated Theorem Proving
Axiom Systems of Present-Day Mathematics
The Model-Theoretic Perspective
Axiom Systems in the Wild
The Topology of Proof Space
Time, Timelessness and Entailment Fabrics
The Notion of Truth
What Can Human Mathematics Be Like?
Going below Axiomatic Mathematics
The Physicalized Laws of Mathematics
Uniformity and Motion in Metamathematical Space
Gravitational and Relativistic Effects in Metamathematics
Empirical Metamathematics
Invented or Discovered? How Mathematics Relates to Humans
What Axioms Can There Be for Human Mathematics?
Counting the Emes of Mathematics and Physics
Some Historical (and Philosophical) Background
Implications for the Future of Mathematics
Hardcover BookNow Available
Originally posted in Stephen Wolfram Writings »
arXiv version »